(*) I'm implicitly assuming that differentiability at $x_0$ implies differentiability at all points around $x_0$ if they are close enough. Only with this assumption I can conclude that the partial derivatives are defined around $x_0$ and therefore ask if they are continuous around $x_0$ or not.

I hope you can follow my thoughts. Else just ask for clarifications, it's my first question.
Thank you for your help :)

$\begingroup$I believe continuity of partial derivatives implies existence of the derivative and hence all directional derivatives. You can see Spivak's 'Calculus on Manifolds' theorem 2.8$\endgroup$
– NL1992Apr 11 at 19:50

$\begingroup$If you assume f has continuous partial derivatives at A open, then it has a continuous differential in A, hence the directional derivatives are also continuous.$\endgroup$
– NL1992Apr 11 at 20:08

$\begingroup$In your proof, to be safe, I would've used the fact f is continuously differentiable.$\endgroup$
– NL1992Apr 11 at 20:10

$\begingroup$You need to tell us more about $X.$$\endgroup$
– zhw.Apr 11 at 20:53